Scaling edge from multiple object with local transform is not working as expected
I think you're confusing several things:


Transformation Orientation will determine what are the directions displayed by the gizmo in the 3D view, and therefore the directions you'll use: are they relative to the Global, Local or custom axis, etc... It has nothing to do with the center of the scale or rotation operations you are going to do.
On the contrary, the Pivot Point will determine what will be the center of your operation when you'll scale or rotate.
In Object mode the object origin will be taken into account if for example you scale an object with Pivot Point > Median Point selected. If you choose Pivot Point > 3D Cursor, it won't, only the 3D cursor will be taken into account.
In Edit mode, the object origin will never be taken into account, if you want this point to be taken into account, put your 3D cursor on it and choose 3D Cursor as pivot point.